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$D(-1)$-triples of triangular numbers

Marija Bliznac Trebješanin

Abstract

We study pairs and triples consisting of triangular numbers such that the product of any two distinct elements decreased by 1 is a perfect square. For a positive integer $n$, we establish a necessary condition for the $n$-th triangular number $T_n$ to be a member of such a pair, and we prove that any such $T_n$ is also a member of infinitely many $D(-1)$-triples.

$D(-1)$-triples of triangular numbers

Abstract

We study pairs and triples consisting of triangular numbers such that the product of any two distinct elements decreased by 1 is a perfect square. For a positive integer , we establish a necessary condition for the -th triangular number to be a member of such a pair, and we prove that any such is also a member of infinitely many -triples.

Paper Structure

This paper contains 4 sections, 7 theorems, 26 equations, 1 table.

Table of Contents

  1. Introduction
  2. Generalized Pellian equation and proof of Theorem \ref{['tm_1']}
  3. $D(-1)$-triples containing $T_n$
  4. Generalization to $D(a)$-triples and future research

Key Result

Theorem 1

Let $n$ be a positive integer such that there exists a positive integer $m$ for which $\{T_n,T_m\}$ is a $D(-1)$-pair. Then both $n$ and $n+1$ are numbers of the form $2^a\cdot p_1^{l_1}\cdots p_k^{l_k}$, where $p_i\equiv 1\pmod{4}$ for each $i\in\{1,\dots,k\}$, $l_1,\dots,l_k$ are positive integers

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Remark
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 3 more